Group-velocity-matched optical parametric oscillatorin tilted quasi-phase-matched gratings

Wei Quan Zhang

An achromatic phase-matching scheme is reported for an optical parametric oscillator in tilted quasi-phase-matched gratings. The spectral angular dispersion is introduced in interaction waves such thateach wave component satisfies the two-dimensional (noncollinear) quasi-phase matching. This is equiv-alent to simultaneous quasi-phase matching and group-velocity matching for ultrashort pulses. Thephase-matching bandwidth for 10 mm periodically poled KTP increases by a factor of 12 at �s � 1.7 �mcompared with one-dimensional quasi-phase matching. The effective interaction length will increase asa result of the matching. © 2006 Optical Society of America

OCIS codes: 190.4410, 190.4970.

1. Introduction

Frequency conversion of ultrashort pulses is impor-tant for many applications, such as communications,signal processing, and spectroscopy. In general, thephase-matching bandwidth is limited by the group-velocity (GV) mismatch between pump and signal (oridler) pulses. Therefore the most common way tobroaden the phase-matching bandwidth is to use athin crystal, which will entail sacrificing conver-sion efficiency. Zhang1–3 has proposed a noncollinearmatching scheme that involves phase matching andGV matching for an optical parametric oscillator(OPO). By using this technique, a wider tunablerange, a larger conversion efficiency, a smaller walk-off effect, and a narrower duration of pulse can beobtained for femtosecond optical parametric genera-tion in KTP.

Quasi-phase matching (QPM) is an attractive tech-nique for frequency conversion and cascade quadraticinteraction because of its large conversion efficiencyand because there is no restriction on the directions ofthe wave vector and polarization. Many authors4,5

have analyzed the characteristics of OPOs with aperiodically poled crystal. Smilgevicius et al.6 demon-strated the operation of a mid-infrared OPO based on

noncollinear interaction in periodically poled KTP(PPKTP). This scheme, however, can suffer frompulse distortions caused by GV-mismatched cascad-ing interactions in a high-conversion-efficiency re-gime. To increase conversion efficiency and decreasepulse distortions, it is significant to have simulta-neous QPM and GV matching in ultrashort-pulseOPOs. Hsu and Yang7 and Zhao et al.8 have discussednoncollinear optical parametric processes on a peri-odically poled crystal, but GV matching is not dis-cussed. Schober et al. have discussed GV-matchedsecond-harmonic generation.9

In this paper I propose an achromatic phase-matching technique for OPOs in a QPM configura-tion, called noncollinear phase matching. In thisscheme I introduce a spectral angular dispersion.Achromatic phase matching is equivalent to GVmatching for an ultrashort-pulse OPO. SimultaneousQPM and GV matching for a QPM OPO at a widerfrequency range can be achieved with the proposedscheme.

This paper is organized as follows: First, the equa-tions that describe achromatic phase-matching con-ditions for a QPM OPO are derived. Second, theequation is applied to PPKTP, and the tuning curve,spectral angular dispersion, phase-matching band-width, and walk-off angle for an ultrashort-pulseOPO are discussed and compared with one-dimensional (1D) QPM.

2. Phase-Matched Tuning Curve in a TiltedQuasi-Phase-Matched Grating

We start with noncollinear phase-matching condi-tions in periodic QPM gratings with a period of �,

The author is with the Department of Physics, Zhejiang Sci-TechUniversity, Hanghou Zhejiang 310033, China. His e-mail addressis mike@hosonic.com.cn.

Received 19 October 2005; revised 3 January 2006; accepted 20January 2006; posted 7 February 2006 (Doc. ID 65473).

0003-6935/06/204977-05$15.00/0© 2006 Optical Society of America

10 July 2006 � Vol. 45, No. 20 � APPLIED OPTICS 4977

tilted from the propagation direction as shown in Fig.1. Here the pump wave and signal (or idler) wavepropagate at angles � and �, respectively, with re-spect to the grating wave vector. The spatial walk-offangle between the pump wave and the signal wave isdenoted as �. In Fig. 1, Kp, Ks, Ki, and K denote thewave vectors of the pump, signal, idler, and QPMgratings, respectively. Waves propagate in the y–zplane perpendicular to the c axis. Also, K � 2���,Kj � 2�nj��j �j � p, s, i�. In the principal axis coordi-nates, the direction cosines of wave vector k are

kx � sin � cos �, ky � sin � sin �, kz � cos �,(1)

where � is the angle between the wave vector k andthe z axis and � is the angle between the projection kin the x–y plane and the x axis. The refractive indicesin the wave vector direction are10

n � 21�2�e � A � �b2 2Bb � A2�1�2�1�2, (2)

where A � kz2c kx

2a, B � kz2c � kx

2a, a � 1�nx2

1�ny2, b � 1�nx

2 1�nz2, c � 1�ny

2 1�nz2, and

e � 1�nx2 � 1�nz

2. The plus and minus signs corre-spond to fast (F) or slow (S) lights, respectively.

A more recent Sellmeier equation11 is

ni2 � Ai � Bi���2 Ci� � Di���2 Ei� �i � x, y, z�,

(3)

where � is in micrometers. Ai, Bi, Ci, Di, and Ei aregiven Table 1. Such noncollinear interaction in atilted QPM grating is described by two kinds of

the wave-vector diagram: Fig. 1(a) corresponds to�s � �p and Fig. 1(b) corresponds to �s � �p � .Assume that all wave vectors lie in y–z plane(� � 90°). The wave-vector mismatch is �K � Kp Ks Ki K. From the geometrical considerations,the projection of the wave-vector mismatch in the zdirection is

�K � Kp cos �p �Ks � Ki�cos��p � K cos � �for Fig. 1�a��, (4a)

or

�K � Kp cos �p �Ks � Ki�cos��p � � K cos � �for Fig. 1�b��, (4b)

where � denotes the angle between the K and z axes.In Fig. 1(a), � � a � �p; in Fig. 1(b), � � a �p. The Taylor expansion of �K��0 � ��� near thepump wavelength of �0 is written as

�K��0 � ��p� � �K��0� � d��K�����d� �0���p�� 1�2 d2��K�����d�2 �0���p�2 � · · · .

(5)

The phase-matching condition obtained from �K��0� � 0is expressed as

��K�2 � K2 Kp2 � �Ks � Ki�2 � 2K�Ks � Ki�cos �

� 0. (6)

Angles � and satisfy

cos � � �K2 � Kp2 �Ks � Ki�2���2KKp�, (7)

cos � �Kp2 � �Ks � Ki�2 K2���2�Ks � Ki�Kp�. (8)

If angles � and are changed, the phase-matchingcondition can be met at a certain range of signalfrequency, and signal tuning is then realized. Let usapply the equations to PPKTP. If the pump wave-length is 526 nm, the phase-matching angle � and thespatial walk-off angle calculated from the achro-matic phase-matching conditions of Eqs. (7) and (8)are plotted in Fig. 2(a) and 2(b) as a function of signalwavelength. The three tuning curves correspond tovalues for two normalized QPM periods. The solid

Table 1. Sellmeier Coefficients for KTP

A B C D E

nx 3.29100 0.04140 0.03978 9.35522 31.45571ny 3.45018 0.04341 0.04597 16.98825 39.43799nz 4.59423 0.06206 0.04763 110.80672 86.12171

Fig. 1. Schematic of a noncollinear OPO in tilted QPM structures:(a) wave-vector diagram for �s � �p � and (b) wave-vector dia-gram for �s � �p � in the y–z plane.

4978 APPLIED OPTICS � Vol. 45, No. 20 � 10 July 2006

curves correspond to � � 15 �m and �s � �p . Thedashed curves correspond to � � 10 �m and �s ��p � . The dashed–dotted curves correspond to� � 10 �m and �s � �p . The phase-matchingangle � and spatial walk-off angle decrease with anincrease of the QPM period �.

3. Spectral Angular Dispersion andGroup-Velocity Matching

By permitting spectral angular dispersion, we caneliminate the first-order term in Eq. (5) to satisfy theGV-matching condition. In fact, d��K�����d� �0

� 0when �K��� �0

� 0 yields

� � d��d�p �i0

� {(Kcos� Kp)dKp�d�p �p0

� (Ks � Ki)[dKs�d�s(d�s�d�p) �s0

� dKi�d�i(d�i�d�p) �i0]}�(KpKsin�), (9)

where ε is the linear spectral angular dispersion inpump pulses required for simultaneous QPM and GVmatching. Assume that d�s�d�p � ��s

2��p2��2 and

d�i�d�p � ��i2��p

2��210:

�� � �d��d�p �0

� [�K cos � � Ks � Ki�[dKs�d�s�d�s�d�p� �s0

� dKi�d�i�d�i�d�p��i0] KpdKp�d�p �p0

]��K�Ks

� Ki�sin ��, (10)

where ε= is the linear spectral angular dispersion insignal (or idler) pulses that one should compensatefor to recombine each frequency component. For thecalculations of dKj�d�j �j � p, s, i�, see Refs. 2 and 12.Spectral angular dispersion � and �� obtained fromEqs. (9) and (10) are plotted in Fig. 3 as a function ofsignal wavelength. From Fig. 3 we can see that spec-tral angular dispersion is larger at � � 15 �m thanthat at � � 10 �m.

4. Phase-Matching Bandwidth

The relative parametric gain is

GR � sinc2����K�2�2 �2�1�2L�, (11)

where L is an interaction length in the crystal and

� � 4� Ep 2deff��ns�s���ni�i�. (12)

Fig. 2. Tuning curves: (a) � angle as a function of signal wave-length �s; (b) � angle as a function of signal wavelength �s. Solidcurves, � � 15 �m and �s � �p �, � � 31.3°. Dashed curves, � �10 �m and �s � �p �, � � 49.4°. Dashed–dotted curves, � � 10 �mand �s � �p �, � � 56.7°.

Fig. 3. Calculated values of the spectral angular dispersion (a) ε=� d��d�p and (b) ε � d��d�p plotted as a function of signal wave-length �s. The solid, dashed, and dashed–dotted curves are thesame as in Fig. 2.

10 July 2006 � Vol. 45, No. 20 � APPLIED OPTICS 4979

Here deff is the effective nonlinear coefficient and Ep isthe electric field intensity of the incident pump light.When GR � GR

max�2, the corresponding phase mis-matching is �K. From sinc2�x� � 1�2, we obtainx � 1.392. Hence

��K� � 2��x�L�2 � �2�1�2. (13)

The phase-matching bandwidth 2��p of the pumpwave is obtained by �K��0 � ���L � 2��x�2 ��2L2�1�2. Without spectral angular dispersion, thephase-matching bandwidth is determined by thefirst-order term in Eq. (5):

���p�L � 2��x�2 � �2L2�1�2�d��K�����d� �0. (14)

When we introduce the spectral angular dispersion,�K has no first-order dependence on �, and the band-width is determined by the second-order term inEq. (5):

���p�2L � 4��x�2 � �2L2�1�2�d2��K�����d�2 �0.

(15)

Figure 4(a) shows ���p�L for 1D QPM calculatedfrom Eq. (14) as a function of signal wavelength. Fig-ure 4(b) shows ���p�2L for noncollinear QPM withlinear spectral angular dispersion, given by Eq. (15).For the example L � 10 mm, Ep � 1.94 � 106 V�mand deff is calculated in Ref. 10. The phase-matchingbandwidths ��p for 1D QPM are 8.07 nm ��s � �p � and 8.3 nm ��s � �p � � at � � 10 �m and�s � 1.7 �m. When there is GV matching, the phase-matching bandwidths for noncollinear QPM are98.3 nm ��s � �p � � and 41.9 nm ��s � �p � at� � 10 �m and �s � 1.7 �m. Therefore the phase-matching bandwidth increases by a factor of 12 whenthe noncollinear QPM configuration is used. Wecan also see that the phase-matching bandwidth at� � 10 �m is approximately three times as large asthat at � � 15 �m for the noncollinear QPM. Fromthese results, achromatic conditions can be adjustedby choice of the grating period.

5. Effective Interaction Length and Walk-Off Effects

Let us now discuss the walk-off effects of Poyntingvectors and spatial walk-off. The walk-off anglebetween the pump and the signal Poynting vectorsis10

�ps � cos1�SpxSsx � SpySsy � SpzSsz�, (16)

where Sij �i � s, p and j � x, y, z� are the directioncosines of the pump and the signal Poynting vectors,respectively. Sij � nr�1�nij

2 1�nr2�kij��1�nij

2 1�n2��n and 1�nr

2 � 1�n2 � n2 ��kij2��1�nij

2 1�n2�2�1. Figure 5 shows the dependence of the walk-off angle of Poynting vectors on signal wavelength.Because the spatial walk-off angle at � � 10 �m is

larger than that at � � 15 �m, the walk-off effect ofPoynting vectors is also larger.

We compare the conversion efficiencies � at 1D andnoncollinear QPM for a 100 fs pulse. Conversion ef-ficiency � is proportional to Lmax

2. Lmax is the maxi-mum effective interactive length, and it can becalculated from the values ��L and ����2L, as shownin Fig. 4, by use of the FWHM 2�� of the spectrum fora 100 fs pulse, which is 35 nm at 1.55 �m. The walk-off angle is smaller at � � 15 �m. Then Lmax for 1DQPM is 2.32 mm at 1.55 �m, whereas that for non-collinear QPM with spectral angular dispersion is15.5 mm at � � 15 �m ��s � �p � and �s � 1.55�m. Because of the square dependence on Lmax, con-version efficiency � for noncollinear QPM increasesby a factor of 50 at 1.55 �m compared with that for1D QPM.

The spatial walk-off that is due to noncollinearQPM and the walk-off of Poynting vectors may de-

Fig. 4. Calculated values of (a) ( �p)L for 1D QPM and (b) ( �p)2Lfor noncollinear QPM with the appropriate spectral angular dis-persion plotted as a function of signal wavelength �s, where ( �p)is the phase-matching bandwidth of the pump wave for L � 10 mm.The solid, dashed, and dashed–dotted curves are the same as inFig. 2.

4980 APPLIED OPTICS � Vol. 45, No. 20 � 10 July 2006

grade the conversion efficiency. The walk-off length isdescribed as La1 � D� and La2 � D��ps, respectively,where D is the beam width. Because the walk-offangle �ps of Poynting vectors is larger than the spatialwalk-off angle , we regard the former. The walk-offangle �ps of Poynting vectors calculated from Eq. (16)is 0.0122 rad at � � 15 �m ��s � �p � and �s �1.55 �m. By equating La2 with Lmax for noncollinearQPM we obtain the critical beam width D to be0.19 mm at 1.55 �m. The Rayleigh angle of 0.19 mmdiameter is 0.00995 rad, and the Rayleigh range is2 cm. The range can match the periodically poledcrystal length �2–3 cm�.

6. Conclusion

An achromatic phase-matching scheme is reportedfor an OPO in which the pump and signal (or idler)waves with appropriate spectral angular dispersionpropagate in tilted QPM gratings. This is equiva-lent to simultaneous QPM and GV matching for ul-trashort pulses. On the basis of a more recentSellmeier formula, the tuning curve and the spectralangular dispersion curve are obtained for PPKTP.The phase-matching bandwidth for noncollinearQPM was shown to increase by a factor of 12 at a

signal wavelength of 1.7 �m, compared with that for1D QPM. In terms of ultrashort pulses, the effectiveinteraction length will increase as a result of GVmatching. The achromatic condition can be adjustedby choice of the grating period. The phase-matchingbandwidth increases with a decrease of the gratingperiod. The calculated walk-off angles show that thewalk-off effect is smaller at � � 15 �m and does notmaintain the advantage of noncollinear QPM. Thistechnique is useful for tunable OPOs with ultrashortpulses in a wide wavelength range.

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Fig. 5. Calculated values of the walk-off angle of Poynting vec-tors plotted as a function of signal wavelength �s. The solid,dashed, and dashed–dotted curves are the same as in Fig. 2.

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